LMFDB - Embedded newform 3100.3.d.h.1301.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 620)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.1
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.h.1301.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.99148i q^{3} -11.3647 q^{7} -15.9149 q^{9} +O(q^{10})$	$q-4.99148i q^{3} -11.3647 q^{7} -15.9149 q^{9} +13.5094i q^{11} +0.886179i q^{13} +12.3209i q^{17} -37.2855 q^{19} +56.7266i q^{21} +24.9525i q^{23} +34.5156i q^{27} -24.1282i q^{29} +(-6.84328 - 30.2352i) q^{31} +67.4320 q^{33} +50.7706i q^{37} +4.42335 q^{39} -32.4591 q^{41} -56.8055i q^{43} +40.4650 q^{47} +80.1559 q^{49} +61.4994 q^{51} -18.1506i q^{53} +186.110i q^{57} +15.8094 q^{59} -61.7600i q^{61} +180.868 q^{63} +38.6570 q^{67} +124.550 q^{69} -98.4074 q^{71} +63.3794i q^{73} -153.530i q^{77} +73.1195i q^{79} +29.0497 q^{81} -95.8136i q^{83} -120.436 q^{87} -162.756i q^{89} -10.0711i q^{91} +(-150.919 + 34.1581i) q^{93} +177.589 q^{97} -215.001i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 44 q^{9}+O(q^{10}) $ 24 * q - 44 * q^9	$ 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) $ 24 * q - 44 * q^9 - 8 * q^19 - 24 * q^31 + 280 * q^39 - 248 * q^41 + 644 * q^49 - 100 * q^51 - 152 * q^59 + 288 * q^69 - 352 * q^71 - 368 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	4.99148i		−	1.66383i	−0.554905	−	0.831914i	$-0.687245\pi$
$3$		−	4.99148i		−	1.66383i	0.554905	−	0.831914i	$-0.312755\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−11.3647			−1.62353			−0.811763	−	0.583987i	$-0.801492\pi$
$7$	−11.3647			−1.62353			−0.811763	+	0.583987i	$0.801492\pi$
$8$		0			0
$9$	−15.9149			−1.76832
$10$		0			0
$11$			13.5094i			1.22813i	0.789256	+	0.614064i	$0.210466\pi$
$11$			13.5094i			1.22813i	−0.789256	+	0.614064i	$0.789534\pi$
$12$		0			0
$13$			0.886179i			0.0681676i	0.999419	+	0.0340838i	$0.0108513\pi$
$13$			0.886179i			0.0681676i	−0.999419	+	0.0340838i	$0.989149\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			12.3209i			0.724758i	0.932031	+	0.362379i	$0.118035\pi$
$17$			12.3209i			0.724758i	−0.932031	+	0.362379i	$0.881965\pi$
$18$		0			0
$19$	−37.2855			−1.96240			−0.981198	−	0.193004i	$-0.938177\pi$
$19$	−37.2855			−1.96240			−0.981198	+	0.193004i	$0.938177\pi$
$20$		0			0
$21$			56.7266i			2.70127i
$22$		0			0
$23$			24.9525i			1.08489i	0.840091	+	0.542446i	$0.182502\pi$
$23$			24.9525i			1.08489i	−0.840091	+	0.542446i	$0.817498\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			34.5156i			1.27835i
$28$		0			0
$29$		−	24.1282i		−	0.832008i	−0.909363	−	0.416004i	$-0.863430\pi$
$29$		−	24.1282i		−	0.832008i	0.909363	−	0.416004i	$-0.136570\pi$
$30$		0			0
$31$	−6.84328	−	30.2352i	−0.220751	−	0.975330i
$31$	−6.84328	−	30.2352i	−0.220751	−	0.975330i
$32$		0			0
$33$	67.4320			2.04339
$34$		0			0
$35$		0			0
$36$		0			0
$37$			50.7706i			1.37218i	0.727517	+	0.686089i	$0.240674\pi$
$37$			50.7706i			1.37218i	−0.727517	+	0.686089i	$0.759326\pi$
$38$		0			0
$39$	4.42335			0.113419
$40$		0			0
$41$	−32.4591			−0.791686			−0.395843	−	0.918318i	$-0.629548\pi$
$41$	−32.4591			−0.791686			−0.395843	+	0.918318i	$0.629548\pi$
$42$		0			0
$43$		−	56.8055i		−	1.32106i	−0.750800	−	0.660529i	$-0.770332\pi$
$43$		−	56.8055i		−	1.32106i	0.750800	−	0.660529i	$-0.229668\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	40.4650			0.860958			0.430479	−	0.902601i	$-0.358345\pi$
$47$	40.4650			0.860958			0.430479	+	0.902601i	$0.358345\pi$
$48$		0			0
$49$	80.1559			1.63583
$50$		0			0
$51$	61.4994			1.20587
$52$		0			0
$53$		−	18.1506i		−	0.342464i	−0.985231	−	0.171232i	$-0.945225\pi$
$53$		−	18.1506i		−	0.342464i	0.985231	−	0.171232i	$-0.0547747\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			186.110i			3.26509i
$58$		0			0
$59$	15.8094			0.267955			0.133978	−	0.990984i	$-0.457225\pi$
$59$	15.8094			0.267955			0.133978	+	0.990984i	$0.457225\pi$
$60$		0			0
$61$		−	61.7600i		−	1.01246i	−0.862399	−	0.506230i	$-0.831039\pi$
$61$		−	61.7600i		−	1.01246i	0.862399	−	0.506230i	$-0.168961\pi$
$62$		0			0
$63$	180.868			2.87091
$64$		0			0
$65$		0			0
$66$		0			0
$67$	38.6570			0.576970			0.288485	−	0.957484i	$-0.406849\pi$
$67$	38.6570			0.576970			0.288485	+	0.957484i	$0.406849\pi$
$68$		0			0
$69$	124.550			1.80507
$70$		0			0
$71$	−98.4074			−1.38602			−0.693010	−	0.720928i	$-0.743716\pi$
$71$	−98.4074			−1.38602			−0.693010	+	0.720928i	$0.743716\pi$
$72$		0			0
$73$			63.3794i			0.868211i	0.900862	+	0.434106i	$0.142936\pi$
$73$			63.3794i			0.868211i	−0.900862	+	0.434106i	$0.857064\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	153.530i		−	1.99390i
$78$		0			0
$79$			73.1195i			0.925563i	0.886472	+	0.462782i	$0.153149\pi$
$79$			73.1195i			0.925563i	−0.886472	+	0.462782i	$0.846851\pi$
$80$		0			0
$81$	29.0497			0.358639
$82$		0			0
$83$		−	95.8136i		−	1.15438i	−0.816610	−	0.577190i	$-0.804149\pi$
$83$		−	95.8136i		−	1.15438i	0.816610	−	0.577190i	$-0.195851\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−120.436			−1.38432
$88$		0			0
$89$		−	162.756i		−	1.82872i	−0.404904	−	0.914359i	$-0.632695\pi$
$89$		−	162.756i		−	1.82872i	0.404904	−	0.914359i	$-0.367305\pi$
$90$		0			0
$91$		−	10.0711i		−	0.110672i
$92$		0			0
$93$	−150.919	+	34.1581i	−1.62278	+	0.367292i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	177.589			1.83082			0.915409	−	0.402524i	$-0.131867\pi$
$97$	177.589			1.83082			0.915409	+	0.402524i	$0.131867\pi$
$98$		0			0
$99$		−	215.001i		−	2.17173i
$100$		0			0
$101$	−98.7784			−0.978004			−0.489002	−	0.872283i	$-0.662639\pi$
$101$	−98.7784			−0.978004			−0.489002	+	0.872283i	$0.662639\pi$
$102$		0			0
$103$	84.1207			0.816706			0.408353	−	0.912824i	$-0.366103\pi$
$103$	84.1207			0.816706			0.408353	+	0.912824i	$0.366103\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.9619			0.121139			0.0605695	−	0.998164i	$-0.480708\pi$
$107$	12.9619			0.121139			0.0605695	+	0.998164i	$0.480708\pi$
$108$		0			0
$109$	−72.0969			−0.661439			−0.330720	−	0.943729i	$-0.607291\pi$
$109$	−72.0969			−0.661439			−0.330720	+	0.943729i	$0.607291\pi$
$110$		0			0
$111$	253.421			2.28307
$112$		0			0
$113$	132.439			1.17203			0.586015	−	0.810300i	$-0.300696\pi$
$113$	132.439			1.17203			0.586015	+	0.810300i	$0.300696\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	14.1034i		−	0.120542i
$118$		0			0
$119$		−	140.023i		−	1.17666i
$120$		0			0
$121$	−61.5041			−0.508299
$122$		0			0
$123$			162.019i			1.31723i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			48.8258i			0.384455i	0.981350	+	0.192228i	$0.0615712\pi$
$127$			48.8258i			0.384455i	−0.981350	+	0.192228i	$0.938429\pi$
$128$		0			0
$129$	−283.544			−2.19801
$130$		0			0
$131$	−20.5254			−0.156683			−0.0783413	−	0.996927i	$-0.524962\pi$
$131$	−20.5254			−0.156683			−0.0783413	+	0.996927i	$0.524962\pi$
$132$		0			0
$133$	423.738			3.18600
$134$		0			0
$135$		0			0
$136$		0			0
$137$			62.5395i			0.456493i	0.973603	+	0.228247i	$0.0732992\pi$
$137$			62.5395i			0.456493i	−0.973603	+	0.228247i	$0.926701\pi$
$138$		0			0
$139$			132.727i			0.954869i	0.878667	+	0.477434i	$0.158433\pi$
$139$			132.727i			0.954869i	−0.878667	+	0.477434i	$0.841567\pi$
$140$		0			0
$141$		−	201.980i		−	1.43249i
$142$		0			0
$143$	−11.9718			−0.0837186
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	400.097i		−	2.72175i
$148$		0			0
$149$	214.171			1.43739			0.718696	−	0.695324i	$-0.244739\pi$
$149$	214.171			1.43739			0.718696	+	0.695324i	$0.244739\pi$
$150$		0			0
$151$		−	59.9860i		−	0.397259i	−0.980075	−	0.198629i	$-0.936351\pi$
$151$		−	59.9860i		−	0.397259i	0.980075	−	0.198629i	$-0.0636490\pi$
$152$		0			0
$153$		−	196.085i		−	1.28160i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	134.780			0.858471			0.429235	−	0.903193i	$-0.358783\pi$
$157$	134.780			0.858471			0.429235	+	0.903193i	$0.358783\pi$
$158$		0			0
$159$	−90.5982			−0.569800
$160$		0			0
$161$		−	283.577i		−	1.76135i
$162$		0			0
$163$	−135.729			−0.832691			−0.416345	−	0.909207i	$-0.636689\pi$
$163$	−135.729			−0.832691			−0.416345	+	0.909207i	$0.636689\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	231.600i		−	1.38683i	−0.720540	−	0.693414i	$-0.756106\pi$
$167$		−	231.600i		−	1.38683i	0.720540	−	0.693414i	$-0.243894\pi$
$168$		0			0
$169$	168.215			0.995353
$170$		0			0
$171$	593.395			3.47015
$172$		0			0
$173$	135.332			0.782268			0.391134	−	0.920334i	$-0.372083\pi$
$173$	135.332			0.782268			0.391134	+	0.920334i	$0.372083\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	78.9122i		−	0.445832i
$178$		0			0
$179$			138.570i			0.774132i	0.922052	+	0.387066i	$0.126511\pi$
$179$			138.570i			0.774132i	−0.922052	+	0.387066i	$0.873489\pi$
$180$		0			0
$181$			230.289i			1.27231i	0.771560	+	0.636157i	$0.219477\pi$
$181$			230.289i			1.27231i	−0.771560	+	0.636157i	$0.780523\pi$
$182$		0			0
$183$	−308.274			−1.68456
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−166.448			−0.890095
$188$		0			0
$189$		−	392.258i		−	2.07544i
$190$		0			0
$191$	−209.621			−1.09749			−0.548747	−	0.835988i	$-0.684895\pi$
$191$	−209.621			−1.09749			−0.548747	+	0.835988i	$0.684895\pi$
$192$		0			0
$193$	−59.3195			−0.307355			−0.153677	−	0.988121i	$-0.549112\pi$
$193$	−59.3195			−0.307355			−0.153677	+	0.988121i	$0.549112\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			38.3616i			0.194729i	0.995249	+	0.0973645i	$0.0310413\pi$
$197$			38.3616i			0.194729i	−0.995249	+	0.0973645i	$0.968959\pi$
$198$		0			0
$199$			26.4630i			0.132980i	0.997787	+	0.0664900i	$0.0211801\pi$
$199$			26.4630i			0.132980i	−0.997787	+	0.0664900i	$0.978820\pi$
$200$		0			0
$201$		−	192.956i		−	0.959978i
$202$		0			0
$203$			274.210i			1.35079i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	397.116i		−	1.91844i
$208$		0			0
$209$		−	503.705i		−	2.41007i
$210$		0			0
$211$	−87.9380			−0.416768			−0.208384	−	0.978047i	$-0.566820\pi$
$211$	−87.9380			−0.416768			−0.208384	+	0.978047i	$0.566820\pi$
$212$		0			0
$213$			491.199i			2.30610i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	77.7717	+	343.614i	0.358395	+	1.58347i
$218$		0			0
$219$	316.357			1.44455
$220$		0			0
$221$	−10.9185			−0.0494050
$222$		0			0
$223$		−	142.523i		−	0.639115i	−0.947567	−	0.319558i	$-0.896466\pi$
$223$		−	142.523i		−	0.639115i	0.947567	−	0.319558i	$-0.103534\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	186.247			0.820473			0.410237	−	0.911979i	$-0.365446\pi$
$227$	186.247			0.820473			0.410237	+	0.911979i	$0.365446\pi$
$228$		0			0
$229$		−	199.863i		−	0.872764i	−0.899761	−	0.436382i	$-0.856260\pi$
$229$		−	199.863i		−	0.872764i	0.899761	−	0.436382i	$-0.143740\pi$
$230$		0			0
$231$	−766.343			−3.31750
$232$		0			0
$233$	−208.256			−0.893804			−0.446902	−	0.894583i	$-0.647473\pi$
$233$	−208.256			−0.893804			−0.446902	+	0.894583i	$0.647473\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	364.975			1.53998
$238$		0			0
$239$			217.736i			0.911031i	0.890228	+	0.455515i	$0.150545\pi$
$239$			217.736i			0.911031i	−0.890228	+	0.455515i	$0.849455\pi$
$240$		0			0
$241$			178.663i			0.741341i	0.928764	+	0.370670i	$0.120872\pi$
$241$			178.663i			0.741341i	−0.928764	+	0.370670i	$0.879128\pi$
$242$		0			0
$243$			165.639i			0.681641i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	33.0417i		−	0.133772i
$248$		0			0
$249$	−478.252			−1.92069
$250$		0			0
$251$			107.233i			0.427222i	0.976919	+	0.213611i	$0.0685225\pi$
$251$			107.233i			0.427222i	−0.976919	+	0.213611i	$0.931477\pi$
$252$		0			0
$253$	−337.094			−1.33239
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−132.303			−0.514796			−0.257398	−	0.966305i	$-0.582865\pi$
$257$	−132.303			−0.514796			−0.257398	+	0.966305i	$0.582865\pi$
$258$		0			0
$259$		−	576.992i		−	2.22777i
$260$		0			0
$261$			383.998i			1.47126i
$262$		0			0
$263$		−	463.215i		−	1.76127i	−0.473792	−	0.880637i	$-0.657115\pi$
$263$		−	463.215i		−	1.76127i	0.473792	−	0.880637i	$-0.342885\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−812.393			−3.04267
$268$		0			0
$269$			413.801i			1.53829i	0.639073	+	0.769146i	$0.279318\pi$
$269$			413.801i			1.53829i	−0.639073	+	0.769146i	$0.720682\pi$
$270$		0			0
$271$			0.867881i			0.00320251i	0.999999	+	0.00160126i	$0.000509696\pi$
$271$			0.867881i			0.00320251i	−0.999999	+	0.00160126i	$0.999490\pi$
$272$		0			0
$273$	−50.2699			−0.184139
$274$		0			0
$275$		0			0
$276$		0			0
$277$			414.804i			1.49749i	0.662859	+	0.748744i	$0.269343\pi$
$277$			414.804i			1.49749i	−0.662859	+	0.748744i	$0.730657\pi$
$278$		0			0
$279$	108.910	+	481.190i	0.390359	+	1.72470i
$280$		0			0
$281$	−61.1955			−0.217778			−0.108889	−	0.994054i	$-0.534729\pi$
$281$	−61.1955			−0.217778			−0.108889	+	0.994054i	$0.534729\pi$
$282$		0			0
$283$	−226.536			−0.800481			−0.400240	−	0.916410i	$-0.631073\pi$
$283$	−226.536			−0.800481			−0.400240	+	0.916410i	$0.631073\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	368.888			1.28532
$288$		0			0
$289$	137.196			0.474727
$290$		0			0
$291$		−	886.434i		−	3.04617i
$292$		0			0
$293$	−453.440			−1.54758			−0.773788	−	0.633444i	$-0.781641\pi$
$293$	−453.440			−1.54758			−0.773788	+	0.633444i	$0.781641\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−466.285			−1.56998
$298$		0			0
$299$	−22.1124			−0.0739545
$300$		0			0
$301$			645.576i			2.14477i
$302$		0			0
$303$			493.051i			1.62723i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−281.766			−0.917805			−0.458903	−	0.888487i	$-0.651757\pi$
$307$	−281.766			−0.917805			−0.458903	+	0.888487i	$0.651757\pi$
$308$		0			0
$309$		−	419.887i		−	1.35886i
$310$		0			0
$311$	252.356			0.811434			0.405717	−	0.913999i	$-0.367022\pi$
$311$	252.356			0.811434			0.405717	+	0.913999i	$0.367022\pi$
$312$		0			0
$313$			288.360i			0.921277i	0.887588	+	0.460639i	$0.152380\pi$
$313$			288.360i			0.921277i	−0.887588	+	0.460639i	$0.847620\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	432.495			1.36434			0.682168	−	0.731195i	$-0.261037\pi$
$317$	432.495			1.36434			0.682168	+	0.731195i	$0.261037\pi$
$318$		0			0
$319$	325.958			1.02181
$320$		0			0
$321$		−	64.6990i		−	0.201554i
$322$		0			0
$323$		−	459.390i		−	1.42226i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			359.870i			1.10052i
$328$		0			0
$329$	−459.872			−1.39779
$330$		0			0
$331$		−	615.665i		−	1.86001i	−0.367542	−	0.930007i	$-0.619801\pi$
$331$		−	615.665i		−	1.86001i	0.367542	−	0.930007i	$-0.380199\pi$
$332$		0			0
$333$		−	808.009i		−	2.42645i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			84.5435i			0.250871i	0.992102	+	0.125436i	$0.0400328\pi$
$337$			84.5435i			0.250871i	−0.992102	+	0.125436i	$0.959967\pi$
$338$		0			0
$339$		−	661.069i		−	1.95006i
$340$		0			0
$341$	408.460	−	92.4487i	1.19783	−	0.271111i
$342$		0			0
$343$	−354.077			−1.03229
$344$		0			0
$345$		0			0
$346$		0			0
$347$			374.447i			1.07910i	0.841954	+	0.539549i	$0.181405\pi$
$347$			374.447i			1.07910i	−0.841954	+	0.539549i	$0.818595\pi$
$348$		0			0
$349$	606.798			1.73868			0.869338	−	0.494219i	$-0.164546\pi$
$349$	606.798			1.73868			0.869338	+	0.494219i	$0.164546\pi$
$350$		0			0
$351$	−30.5870			−0.0871424
$352$		0			0
$353$		−	166.830i		−	0.472607i	−0.971679	−	0.236304i	$-0.924064\pi$
$353$		−	166.830i		−	0.472607i	0.971679	−	0.236304i	$-0.0759360\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−698.921			−1.95776
$358$		0			0
$359$	216.147			0.602081			0.301041	−	0.953611i	$-0.402666\pi$
$359$	216.147			0.602081			0.301041	+	0.953611i	$0.402666\pi$
$360$		0			0
$361$	1029.21			2.85100
$362$		0			0
$363$			306.997i			0.845721i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	572.502i		−	1.55995i	−0.625810	−	0.779975i	$-0.715232\pi$
$367$		−	572.502i		−	1.55995i	0.625810	−	0.779975i	$-0.284768\pi$
$368$		0			0
$369$	516.584			1.39996
$370$		0			0
$371$			206.275i			0.555998i
$372$		0			0
$373$	443.172			1.18813			0.594064	−	0.804418i	$-0.297522\pi$
$373$	443.172			1.18813			0.594064	+	0.804418i	$0.297522\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.3819			0.0567160
$378$		0			0
$379$	−43.2994			−0.114246			−0.0571232	−	0.998367i	$-0.518193\pi$
$379$	−43.2994			−0.114246			−0.0571232	+	0.998367i	$0.518193\pi$
$380$		0			0
$381$	243.713			0.639667
$382$		0			0
$383$		−	286.780i		−	0.748773i	−0.927273	−	0.374386i	$-0.877853\pi$
$383$		−	286.780i		−	0.748773i	0.927273	−	0.374386i	$-0.122147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			904.053i			2.33606i
$388$		0			0
$389$			475.452i			1.22224i	0.791537	+	0.611121i	$0.209281\pi$
$389$			475.452i			1.22224i	−0.791537	+	0.611121i	$0.790719\pi$
$390$		0			0
$391$	−307.437			−0.786283
$392$		0			0
$393$			102.452i			0.260693i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−143.558			−0.361607			−0.180804	−	0.983519i	$-0.557870\pi$
$397$	−143.558			−0.361607			−0.180804	+	0.983519i	$0.557870\pi$
$398$		0			0
$399$		−	2115.08i		−	5.30095i
$400$		0			0
$401$		−	434.489i		−	1.08351i	−0.840535	−	0.541757i	$-0.817759\pi$
$401$		−	434.489i		−	1.08351i	0.840535	−	0.541757i	$-0.182241\pi$
$402$		0			0
$403$	26.7938	−	6.06438i	0.0664860	−	0.0150481i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−685.881			−1.68521
$408$		0			0
$409$		−	421.472i		−	1.03049i	−0.857042	−	0.515247i	$-0.827700\pi$
$409$		−	421.472i		−	1.03049i	0.857042	−	0.515247i	$-0.172300\pi$
$410$		0			0
$411$	312.165			0.759526
$412$		0			0
$413$	−179.668			−0.435032
$414$		0			0
$415$		0			0
$416$		0			0
$417$	662.503			1.58874
$418$		0			0
$419$	−203.529			−0.485750			−0.242875	−	0.970058i	$-0.578090\pi$
$419$	−203.529			−0.485750			−0.242875	+	0.970058i	$0.578090\pi$
$420$		0			0
$421$	585.993			1.39191			0.695953	−	0.718087i	$-0.254982\pi$
$421$	585.993			1.39191			0.695953	+	0.718087i	$0.254982\pi$
$422$		0			0
$423$	−643.996			−1.52245
$424$		0			0
$425$		0			0
$426$		0			0
$427$			701.883i			1.64375i
$428$		0			0
$429$			59.7568i			0.139293i
$430$		0			0
$431$	−250.275			−0.580685			−0.290343	−	0.956923i	$-0.593769\pi$
$431$	−250.275			−0.580685			−0.290343	+	0.956923i	$0.593769\pi$
$432$		0			0
$433$		−	557.133i		−	1.28668i	−0.765580	−	0.643340i	$-0.777548\pi$
$433$		−	557.133i		−	1.28668i	0.765580	−	0.643340i	$-0.222452\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	930.367i		−	2.12899i
$438$		0			0
$439$	−393.661			−0.896722			−0.448361	−	0.893853i	$-0.647992\pi$
$439$	−393.661			−0.896722			−0.448361	+	0.893853i	$0.647992\pi$
$440$		0			0
$441$	−1275.67			−2.89268
$442$		0			0
$443$	−267.533			−0.603913			−0.301956	−	0.953322i	$-0.597640\pi$
$443$	−267.533			−0.603913			−0.301956	+	0.953322i	$0.597640\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	1069.03i		−	2.39157i
$448$		0			0
$449$			500.447i			1.11458i	0.830317	+	0.557291i	$0.188159\pi$
$449$			500.447i			1.11458i	−0.830317	+	0.557291i	$0.811841\pi$
$450$		0			0
$451$		−	438.504i		−	0.972292i
$452$		0			0
$453$	−299.419			−0.660970
$454$		0			0
$455$		0			0
$456$		0			0
$457$			465.166i			1.01787i	0.860805	+	0.508935i	$0.169960\pi$
$457$			465.166i			1.01787i	−0.860805	+	0.508935i	$0.830040\pi$
$458$		0			0
$459$	−425.262			−0.926497
$460$		0			0
$461$			230.870i			0.500802i	0.968142	+	0.250401i	$0.0805624\pi$
$461$			230.870i			0.500802i	−0.968142	+	0.250401i	$0.919438\pi$
$462$		0			0
$463$			265.858i			0.574208i	0.957899	+	0.287104i	$0.0926925\pi$
$463$			265.858i			0.574208i	−0.957899	+	0.287104i	$0.907307\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−622.574			−1.33313			−0.666567	−	0.745445i	$-0.732237\pi$
$467$	−622.574			−1.33313			−0.666567	+	0.745445i	$0.732237\pi$
$468$		0			0
$469$	−439.324			−0.936725
$470$		0			0
$471$		−	672.751i		−	1.42835i
$472$		0			0
$473$	767.409			1.62243
$474$		0			0
$475$		0			0
$476$		0			0
$477$			288.864i			0.605586i
$478$		0			0
$479$	226.221			0.472277			0.236139	−	0.971719i	$-0.424118\pi$
$479$	226.221			0.472277			0.236139	+	0.971719i	$0.424118\pi$
$480$		0			0
$481$	−44.9919			−0.0935382
$482$		0			0
$483$	−1415.47			−2.93058
$484$		0			0
$485$		0			0
$486$		0			0
$487$			269.408i			0.553199i	0.960985	+	0.276600i	$0.0892076\pi$
$487$			269.408i			0.553199i	−0.960985	+	0.276600i	$0.910792\pi$
$488$		0			0
$489$			677.487i			1.38545i
$490$		0			0
$491$		−	132.546i		−	0.269952i	−0.990849	−	0.134976i	$-0.956904\pi$
$491$		−	132.546i		−	0.269952i	0.990849	−	0.134976i	$-0.0430957\pi$
$492$		0			0
$493$	297.281			0.603004
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1118.37			2.25024
$498$		0			0
$499$		−	913.312i		−	1.83029i	−0.403130	−	0.915143i	$-0.632078\pi$
$499$		−	913.312i		−	1.83029i	0.403130	−	0.915143i	$-0.367922\pi$
$500$		0			0
$501$	−1156.03			−2.30744
$502$		0			0
$503$	815.846			1.62196			0.810980	−	0.585073i	$-0.198934\pi$
$503$	815.846			1.62196			0.810980	+	0.585073i	$0.198934\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	839.641i		−	1.65610i
$508$		0			0
$509$		−	249.141i		−	0.489472i	−0.969590	−	0.244736i	$-0.921299\pi$
$509$		−	249.141i		−	0.489472i	0.969590	−	0.244736i	$-0.0787013\pi$
$510$		0			0
$511$		−	720.287i		−	1.40956i
$512$		0			0
$513$		−	1286.93i		−	2.50864i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			546.659i			1.05737i
$518$		0			0
$519$		−	675.509i		−	1.30156i
$520$		0			0
$521$	479.982			0.921270			0.460635	−	0.887590i	$-0.347622\pi$
$521$	479.982			0.921270			0.460635	+	0.887590i	$0.347622\pi$
$522$		0			0
$523$		−	563.365i		−	1.07718i	−0.842568	−	0.538590i	$-0.818957\pi$
$523$		−	563.365i		−	1.07718i	0.842568	−	0.538590i	$-0.181043\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	372.525	−	84.3152i	0.706878	−	0.159991i
$528$		0			0
$529$	−93.6272			−0.176989
$530$		0			0
$531$	−251.604			−0.473831
$532$		0			0
$533$		−	28.7646i		−	0.0539674i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	691.668			1.28802
$538$		0			0
$539$			1082.86i			2.00901i
$540$		0			0
$541$	−478.035			−0.883613			−0.441806	−	0.897110i	$-0.645662\pi$
$541$	−478.035			−0.883613			−0.441806	+	0.897110i	$0.645662\pi$
$542$		0			0
$543$	1149.48			2.11691
$544$		0			0
$545$		0			0
$546$		0			0
$547$	332.260			0.607423			0.303711	−	0.952764i	$-0.401774\pi$
$547$	332.260			0.607423			0.303711	+	0.952764i	$0.401774\pi$
$548$		0			0
$549$			982.904i			1.79035i
$550$		0			0
$551$			899.634i			1.63273i
$552$		0			0
$553$		−	830.980i		−	1.50268i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	559.956i		−	1.00531i	−0.864488	−	0.502653i	$-0.832357\pi$
$557$		−	559.956i		−	1.00531i	0.864488	−	0.502653i	$-0.167643\pi$
$558$		0			0
$559$	50.3399			0.0900534
$560$		0			0
$561$			830.821i			1.48096i
$562$		0			0
$563$	712.196			1.26500			0.632501	−	0.774560i	$-0.282028\pi$
$563$	712.196			1.26500			0.632501	+	0.774560i	$0.282028\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−330.141			−0.582259
$568$		0			0
$569$		−	707.768i		−	1.24388i	−0.783065	−	0.621940i	$-0.786345\pi$
$569$		−	707.768i		−	1.24388i	0.783065	−	0.621940i	$-0.213655\pi$
$570$		0			0
$571$			999.196i			1.74991i	0.484208	+	0.874953i	$0.339108\pi$
$571$			999.196i			1.74991i	−0.484208	+	0.874953i	$0.660892\pi$
$572$		0			0
$573$			1046.32i			1.82604i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−55.3022			−0.0958443			−0.0479221	−	0.998851i	$-0.515260\pi$
$577$	−55.3022			−0.0958443			−0.0479221	+	0.998851i	$0.515260\pi$
$578$		0			0
$579$			296.092i			0.511385i
$580$		0			0
$581$			1088.89i			1.87417i
$582$		0			0
$583$	245.203			0.420589
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	379.414i		−	0.646361i	−0.946337	−	0.323180i	$-0.895248\pi$
$587$		−	379.414i		−	0.646361i	0.946337	−	0.323180i	$-0.104752\pi$
$588$		0			0
$589$	255.155	+	1127.34i	0.433201	+	1.91398i
$590$		0			0
$591$	191.481			0.323996
$592$		0			0
$593$	614.685			1.03657			0.518284	−	0.855209i	$-0.326571\pi$
$593$	614.685			1.03657			0.518284	+	0.855209i	$0.326571\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	132.090			0.221256
$598$		0			0
$599$	−370.752			−0.618951			−0.309475	−	0.950907i	$-0.600153\pi$
$599$	−370.752			−0.618951			−0.309475	+	0.950907i	$0.600153\pi$
$600$		0			0
$601$		−	112.901i		−	0.187856i	−0.995579	−	0.0939278i	$-0.970058\pi$
$601$		−	112.901i		−	0.187856i	0.995579	−	0.0939278i	$-0.0299423\pi$
$602$		0			0
$603$	−615.221			−1.02027
$604$		0			0
$605$		0			0
$606$		0			0
$607$	178.484			0.294042			0.147021	−	0.989133i	$-0.453031\pi$
$607$	178.484			0.294042			0.147021	+	0.989133i	$0.453031\pi$
$608$		0			0
$609$	1368.71			2.24747
$610$		0			0
$611$			35.8593i			0.0586895i
$612$		0			0
$613$			35.8214i			0.0584362i	0.999573	+	0.0292181i	$0.00930174\pi$
$613$			35.8214i			0.0584362i	−0.999573	+	0.0292181i	$0.990698\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	936.451			1.51775			0.758874	−	0.651237i	$-0.225750\pi$
$617$	936.451			1.51775			0.758874	+	0.651237i	$0.225750\pi$
$618$		0			0
$619$			42.6845i			0.0689573i	0.999405	+	0.0344786i	$0.0109771\pi$
$619$			42.6845i			0.0689573i	−0.999405	+	0.0344786i	$0.989023\pi$
$620$		0			0
$621$	−861.249			−1.38687
$622$		0			0
$623$			1849.67i			2.96897i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−2514.24			−4.00995
$628$		0			0
$629$	−625.539			−0.994497
$630$		0			0
$631$			168.724i			0.267391i	0.991022	+	0.133695i	$0.0426844\pi$
$631$			168.724i			0.267391i	−0.991022	+	0.133695i	$0.957316\pi$
$632$		0			0
$633$			438.941i			0.693429i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			71.0325i			0.111511i
$638$		0			0
$639$	1566.14			2.45093
$640$		0			0
$641$			927.157i			1.44642i	0.690627	+	0.723212i	$0.257335\pi$
$641$			927.157i			1.44642i	−0.690627	+	0.723212i	$0.742665\pi$
$642$		0			0
$643$			838.434i			1.30394i	0.758245	+	0.651970i	$0.226057\pi$
$643$			838.434i			1.30394i	−0.758245	+	0.651970i	$0.773943\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			707.085i			1.09287i	0.837502	+	0.546434i	$0.184015\pi$
$647$			707.085i			1.09287i	−0.837502	+	0.546434i	$0.815985\pi$
$648$		0			0
$649$			213.575i			0.329084i
$650$		0			0
$651$	1715.14	−	388.196i	2.63463	−	0.596307i
$652$		0			0
$653$	419.754			0.642809			0.321404	−	0.946942i	$-0.395845\pi$
$653$	419.754			0.642809			0.321404	+	0.946942i	$0.395845\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	1008.68i		−	1.53528i
$658$		0			0
$659$	120.524			0.182890			0.0914450	−	0.995810i	$-0.470851\pi$
$659$	120.524			0.182890			0.0914450	+	0.995810i	$0.470851\pi$
$660$		0			0
$661$	315.011			0.476567			0.238283	−	0.971196i	$-0.423415\pi$
$661$	315.011			0.476567			0.238283	+	0.971196i	$0.423415\pi$
$662$		0			0
$663$			54.4995i			0.0822014i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	602.060			0.902638
$668$		0			0
$669$	−711.399			−1.06338
$670$		0			0
$671$	834.342			1.24343
$672$		0			0
$673$			386.080i			0.573670i	0.957980	+	0.286835i	$0.0926032\pi$
$673$			386.080i			0.573670i	−0.957980	+	0.286835i	$0.907397\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	832.627i		−	1.22988i	−0.788575	−	0.614938i	$-0.789181\pi$
$677$		−	832.627i		−	1.22988i	0.788575	−	0.614938i	$-0.210819\pi$
$678$		0			0
$679$	−2018.25			−2.97238
$680$		0			0
$681$		−	929.650i		−	1.36513i
$682$		0			0
$683$	−438.615			−0.642190			−0.321095	−	0.947047i	$-0.604051\pi$
$683$	−438.615			−0.642190			−0.321095	+	0.947047i	$0.604051\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−997.613			−1.45213
$688$		0			0
$689$	16.0847			0.0233449
$690$		0			0
$691$	482.753			0.698630			0.349315	−	0.937005i	$-0.386414\pi$
$691$	482.753			0.698630			0.349315	+	0.937005i	$0.386414\pi$
$692$		0			0
$693$			2443.41i			3.52585i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	399.925i		−	0.573781i
$698$		0			0
$699$			1039.51i			1.48714i
$700$		0			0
$701$	−115.415			−0.164644			−0.0823220	−	0.996606i	$-0.526234\pi$
$701$	−115.415			−0.164644			−0.0823220	+	0.996606i	$0.526234\pi$
$702$		0			0
$703$		−	1893.01i		−	2.69276i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1122.58			1.58781
$708$		0			0
$709$		−	627.935i		−	0.885663i	−0.896605	−	0.442831i	$-0.853974\pi$
$709$		−	627.935i		−	0.885663i	0.896605	−	0.442831i	$-0.146026\pi$
$710$		0			0
$711$		−	1163.69i		−	1.63669i
$712$		0			0
$713$	754.445	−	170.757i	1.05813	−	0.239491i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1086.83			1.51580
$718$		0			0
$719$		−	506.896i		−	0.705002i	−0.935811	−	0.352501i	$-0.885331\pi$
$719$		−	506.896i		−	0.705002i	0.935811	−	0.352501i	$-0.114669\pi$
$720$		0			0
$721$	−956.005			−1.32594
$722$		0			0
$723$	891.794			1.23346
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1142.91			1.57209			0.786045	−	0.618169i	$-0.212125\pi$
$727$	1142.91			1.57209			0.786045	+	0.618169i	$0.212125\pi$
$728$		0			0
$729$	1088.23			1.49277
$730$		0			0
$731$	699.894			0.957447
$732$		0			0
$733$	362.368			0.494363			0.247182	−	0.968969i	$-0.420496\pi$
$733$	362.368			0.494363			0.247182	+	0.968969i	$0.420496\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			522.233i			0.708593i
$738$		0			0
$739$			225.656i			0.305354i	0.988276	+	0.152677i	$0.0487894\pi$
$739$			225.656i			0.305354i	−0.988276	+	0.152677i	$0.951211\pi$
$740$		0			0
$741$	−164.927			−0.222573
$742$		0			0
$743$		−	888.886i		−	1.19635i	−0.801367	−	0.598173i	$-0.795893\pi$
$743$		−	888.886i		−	1.19635i	0.801367	−	0.598173i	$-0.204107\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			1524.86i			2.04132i
$748$		0			0
$749$	−147.308			−0.196672
$750$		0			0
$751$	884.452			1.17770			0.588849	−	0.808243i	$-0.299581\pi$
$751$	884.452			1.17770			0.588849	+	0.808243i	$0.299581\pi$
$752$		0			0
$753$	535.250			0.710824
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	1096.55i		−	1.44855i	−0.689513	−	0.724274i	$-0.742175\pi$
$757$		−	1096.55i		−	1.44855i	0.689513	−	0.724274i	$-0.257825\pi$
$758$		0			0
$759$			1682.60i			2.21686i
$760$		0			0
$761$		−	1061.13i		−	1.39439i	−0.716883	−	0.697193i	$-0.754432\pi$
$761$		−	1061.13i		−	1.39439i	0.716883	−	0.697193i	$-0.245568\pi$
$762$		0			0
$763$	819.358			1.07386
$764$		0			0
$765$		0			0
$766$		0			0
$767$			14.0099i			0.0182659i
$768$		0			0
$769$	−244.794			−0.318328			−0.159164	−	0.987252i	$-0.550880\pi$
$769$	−244.794			−0.318328			−0.159164	+	0.987252i	$0.550880\pi$
$770$		0			0
$771$			660.386i			0.856532i
$772$		0			0
$773$			339.715i			0.439476i	0.975559	+	0.219738i	$0.0705203\pi$
$773$			339.715i			0.439476i	−0.975559	+	0.219738i	$0.929480\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−2880.04			−3.70662
$778$		0			0
$779$	1210.26			1.55360
$780$		0			0
$781$		−	1329.43i		−	1.70221i
$782$		0			0
$783$	832.799			1.06360
$784$		0			0
$785$		0			0
$786$		0			0
$787$			514.755i			0.654072i	0.945012	+	0.327036i	$0.106050\pi$
$787$			514.755i			0.654072i	−0.945012	+	0.327036i	$0.893950\pi$
$788$		0			0
$789$	−2312.13			−2.93045
$790$		0			0
$791$	−1505.13			−1.90282
$792$		0			0
$793$	54.7305			0.0690170
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	180.242i		−	0.226150i	−0.993586	−	0.113075i	$-0.963930\pi$
$797$		−	180.242i		−	0.226150i	0.993586	−	0.113075i	$-0.0360701\pi$
$798$		0			0
$799$			498.565i			0.623986i
$800$		0			0
$801$			2590.24i			3.23376i
$802$		0			0
$803$	−856.219			−1.06627
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2065.48			2.55945
$808$		0			0
$809$		−	351.937i		−	0.435028i	−0.976057	−	0.217514i	$-0.930205\pi$
$809$		−	351.937i		−	0.435028i	0.976057	−	0.217514i	$-0.0697947\pi$
$810$		0			0
$811$	−1227.62			−1.51372			−0.756858	−	0.653580i	$-0.773266\pi$
$811$	−1227.62			−1.51372			−0.756858	+	0.653580i	$0.773266\pi$
$812$		0			0
$813$	4.33201			0.00532843
$814$		0			0
$815$		0			0
$816$		0			0
$817$			2118.02i			2.59244i
$818$		0			0
$819$			160.281i			0.195703i
$820$		0			0
$821$			119.198i			0.145187i	0.997362	+	0.0725934i	$0.0231275\pi$
$821$			119.198i			0.145187i	−0.997362	+	0.0725934i	$0.976872\pi$
$822$		0			0
$823$			1310.76i			1.59266i	0.604866	+	0.796328i	$0.293227\pi$
$823$			1310.76i			1.59266i	−0.604866	+	0.796328i	$0.706773\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			449.357i			0.543358i	0.962388	+	0.271679i	$0.0875788\pi$
$827$			449.357i			0.543358i	−0.962388	+	0.271679i	$0.912421\pi$
$828$		0			0
$829$			553.749i			0.667973i	0.942578	+	0.333986i	$0.108394\pi$
$829$			553.749i			0.667973i	−0.942578	+	0.333986i	$0.891606\pi$
$830$		0			0
$831$	2070.49			2.49156
$832$		0			0
$833$			987.591i			1.18558i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1043.59	−	236.200i	1.24682	−	0.282198i
$838$		0			0
$839$	−170.709			−0.203467			−0.101734	−	0.994812i	$-0.532439\pi$
$839$	−170.709			−0.203467			−0.101734	+	0.994812i	$0.532439\pi$
$840$		0			0
$841$	258.828			0.307763
$842$		0			0
$843$			305.456i			0.362344i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	698.975			0.825236
$848$		0			0
$849$			1130.75i			1.33186i
$850$		0			0
$851$	−1266.85			−1.48866
$852$		0			0
$853$	479.850			0.562544			0.281272	−	0.959628i	$-0.409244\pi$
$853$	479.850			0.562544			0.281272	+	0.959628i	$0.409244\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−165.299			−0.192881			−0.0964403	−	0.995339i	$-0.530746\pi$
$857$	−165.299			−0.192881			−0.0964403	+	0.995339i	$0.530746\pi$
$858$		0			0
$859$		−	1155.66i		−	1.34535i	−0.739937	−	0.672676i	$-0.765145\pi$
$859$		−	1155.66i		−	1.34535i	0.739937	−	0.672676i	$-0.234855\pi$
$860$		0			0
$861$		−	1841.30i		−	2.13856i
$862$		0			0
$863$			472.551i			0.547567i	0.961791	+	0.273784i	$0.0882753\pi$
$863$			472.551i			0.547567i	−0.961791	+	0.273784i	$0.911725\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	684.811i		−	0.789863i
$868$		0			0
$869$	−987.801			−1.13671
$870$		0			0
$871$			34.2570i			0.0393307i
$872$		0			0
$873$	−2826.32			−3.23748
$874$		0			0
$875$		0			0
$876$		0			0
$877$	163.272			0.186171			0.0930854	−	0.995658i	$-0.470327\pi$
$877$	163.272			0.186171			0.0930854	+	0.995658i	$0.470327\pi$
$878$		0			0
$879$			2263.34i			2.57490i
$880$		0			0
$881$		−	119.672i		−	0.135837i	−0.997691	−	0.0679185i	$-0.978364\pi$
$881$		−	119.672i		−	0.135837i	0.997691	−	0.0679185i	$-0.0216358\pi$
$882$		0			0
$883$			876.902i			0.993093i	0.868010	+	0.496547i	$0.165399\pi$
$883$			876.902i			0.993093i	−0.868010	+	0.496547i	$0.834601\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	154.150			0.173788			0.0868941	−	0.996218i	$-0.472306\pi$
$887$	154.150			0.173788			0.0868941	+	0.996218i	$0.472306\pi$
$888$		0			0
$889$		−	554.890i		−	0.624173i
$890$		0			0
$891$			392.445i			0.440454i
$892$		0			0
$893$	−1508.76			−1.68954
$894$		0			0
$895$		0			0
$896$		0			0
$897$			110.374i			0.123047i
$898$		0			0
$899$	−729.523	+	165.116i	−0.811483	+	0.183667i
$900$		0			0
$901$	223.631			0.248203
$902$		0			0
$903$	3222.38			3.56853
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−93.4111			−0.102989			−0.0514945	−	0.998673i	$-0.516398\pi$
$907$	−93.4111			−0.102989			−0.0514945	+	0.998673i	$0.516398\pi$
$908$		0			0
$909$	1572.05			1.72943
$910$		0			0
$911$		−	659.085i		−	0.723475i	−0.932280	−	0.361737i	$-0.882184\pi$
$911$		−	659.085i		−	0.723475i	0.932280	−	0.361737i	$-0.117816\pi$
$912$		0			0
$913$	1294.38			1.41773
$914$		0			0
$915$		0			0
$916$		0			0
$917$	233.265			0.254378
$918$		0			0
$919$	732.440			0.796997			0.398498	−	0.917169i	$-0.369531\pi$
$919$	732.440			0.796997			0.398498	+	0.917169i	$0.369531\pi$
$920$		0			0
$921$			1406.43i			1.52707i
$922$		0			0
$923$		−	87.2066i		−	0.0944817i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1338.77			−1.44420
$928$		0			0
$929$			38.0696i			0.0409791i	0.999790	+	0.0204895i	$0.00652248\pi$
$929$			38.0696i			0.0409791i	−0.999790	+	0.0204895i	$0.993478\pi$
$930$		0			0
$931$	−2988.65			−3.21016
$932$		0			0
$933$		−	1259.63i		−	1.35009i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−900.249			−0.960778			−0.480389	−	0.877056i	$-0.659504\pi$
$937$	−900.249			−0.960778			−0.480389	+	0.877056i	$0.659504\pi$
$938$		0			0
$939$	1439.34			1.53285
$940$		0			0
$941$			1304.27i			1.38605i	0.720915	+	0.693024i	$0.243722\pi$
$941$			1304.27i			1.38605i	−0.720915	+	0.693024i	$0.756278\pi$
$942$		0			0
$943$		−	809.937i		−	0.858894i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	478.837i		−	0.505636i	−0.967514	−	0.252818i	$-0.918643\pi$
$947$		−	478.837i		−	0.505636i	0.967514	−	0.252818i	$-0.0813574\pi$
$948$		0			0
$949$	−56.1655			−0.0591839
$950$		0			0
$951$		−	2158.79i		−	2.27002i
$952$		0			0
$953$		−	472.273i		−	0.495565i	−0.968816	−	0.247782i	$-0.920298\pi$
$953$		−	472.273i		−	0.495565i	0.968816	−	0.247782i	$-0.0797018\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	1627.01i		−	1.70012i
$958$		0			0
$959$		−	710.742i		−	0.741128i
$960$		0			0
$961$	−867.339	+	413.816i	−0.902538	+	0.430610i
$962$		0			0
$963$	−206.287			−0.214213
$964$		0			0
$965$		0			0
$966$		0			0
$967$			74.4071i			0.0769463i	0.999260	+	0.0384732i	$0.0122494\pi$
$967$			74.4071i			0.0769463i	−0.999260	+	0.0384732i	$0.987751\pi$
$968$		0			0
$969$	−2293.04			−2.36640
$970$		0			0
$971$	−633.355			−0.652271			−0.326135	−	0.945323i	$-0.605747\pi$
$971$	−633.355			−0.652271			−0.326135	+	0.945323i	$0.605747\pi$
$972$		0			0
$973$		−	1508.40i		−	1.55025i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1417.47			−1.45084			−0.725420	−	0.688306i	$-0.758354\pi$
$977$	−1417.47			−1.45084			−0.725420	+	0.688306i	$0.758354\pi$
$978$		0			0
$979$	2198.74			2.24590
$980$		0			0
$981$	1147.41			1.16964
$982$		0			0
$983$			364.691i			0.370998i	0.982644	+	0.185499i	$0.0593901\pi$
$983$			364.691i			0.370998i	−0.982644	+	0.185499i	$0.940610\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			2295.44i			2.32568i
$988$		0			0
$989$	1417.44			1.43320
$990$		0			0
$991$		−	182.515i		−	0.184173i	−0.995751	−	0.0920863i	$-0.970646\pi$
$991$		−	182.515i		−	0.184173i	0.995751	−	0.0920863i	$-0.0293536\pi$
$992$		0			0
$993$	−3073.08			−3.09474
$994$		0			0
$995$		0			0
$996$		0			0
$997$	854.935			0.857508			0.428754	−	0.903421i	$-0.358953\pi$
$997$	854.935			0.857508			0.428754	+	0.903421i	$0.358953\pi$
$998$		0			0
$999$	−1752.38			−1.75413

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.h.1301.1		24
5.2	odd	4		620.3.f.c.309.2	yes	24
5.3	odd	4		620.3.f.c.309.23	yes	24
5.4	even	2	inner	3100.3.d.h.1301.24		24
31.30	odd	2	inner	3100.3.d.h.1301.2		24
155.92	even	4		620.3.f.c.309.24	yes	24
155.123	even	4		620.3.f.c.309.1	✓	24
155.154	odd	2	inner	3100.3.d.h.1301.23		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
620.3.f.c.309.1	✓	24	155.123	even	4
620.3.f.c.309.2	yes	24	5.2	odd	4
620.3.f.c.309.23	yes	24	5.3	odd	4
620.3.f.c.309.24	yes	24	155.92	even	4
3100.3.d.h.1301.1		24	1.1	even	1	trivial
3100.3.d.h.1301.2		24	31.30	odd	2	inner
3100.3.d.h.1301.23		24	155.154	odd	2	inner
3100.3.d.h.1301.24		24	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.99148i q^{3} -11.3647 q^{7} -15.9149 q^{9} +O(q^{10})\)	\(q-4.99148i q^{3} -11.3647 q^{7} -15.9149 q^{9} +13.5094i q^{11} +0.886179i q^{13} +12.3209i q^{17} -37.2855 q^{19} +56.7266i q^{21} +24.9525i q^{23} +34.5156i q^{27} -24.1282i q^{29} +(-6.84328 - 30.2352i) q^{31} +67.4320 q^{33} +50.7706i q^{37} +4.42335 q^{39} -32.4591 q^{41} -56.8055i q^{43} +40.4650 q^{47} +80.1559 q^{49} +61.4994 q^{51} -18.1506i q^{53} +186.110i q^{57} +15.8094 q^{59} -61.7600i q^{61} +180.868 q^{63} +38.6570 q^{67} +124.550 q^{69} -98.4074 q^{71} +63.3794i q^{73} -153.530i q^{77} +73.1195i q^{79} +29.0497 q^{81} -95.8136i q^{83} -120.436 q^{87} -162.756i q^{89} -10.0711i q^{91} +(-150.919 + 34.1581i) q^{93} +177.589 q^{97} -215.001i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 44 q^{9}+O(q^{10}) \) 24 * q - 44 * q^9	\( 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) \) 24 * q - 44 * q^9 - 8 * q^19 - 24 * q^31 + 280 * q^39 - 248 * q^41 + 644 * q^49 - 100 * q^51 - 152 * q^59 + 288 * q^69 - 352 * q^71 - 368 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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